Theorems

Little sites

Idea

Let CC be a category with a pretopologyJJ (i.e. a site) and aa an object of CC. As an analogy with sheaves on a topological space XX, which are defined on the site Op(X)Op(X) of open sets of XX, we can try to define sheaves on aa, using the elements of covering families of aa from JJ. This is called the little site of aa, in contrast to the big site of aa which is the slice category C/aC/a with its induced topology.

A little site may sometimes be called a small site, but it's probably best to save that name for a site which is a small category.

Definition

David Roberts: The following is experimental, use at own risk, although I’m sure it has been thought about before.

Consider the subcategory J/aJ/a of C/aC/a with objects u0→au_0 \to a such that u0u_0 is a member of some covering family U={ui→a}U = \{u_i \to a\}. Given two such objects u0→au_0 \to a, v0→av_0 \to a, and covering families UU, VV that contain them, there is a covering family W=UVW = UV which is the pullback (or at least a weak pullback) of UU and VV in CC. There is then some element ww of WW such that there is a square

such that v0→u0v_0 \to u_0 is an element of a covering family of u0u_0, so the arrows w→u0w \to u_0 and w→v0w \to v_0 really are morphisms of J/aJ/a. Then we say a covering family of u0→au_0\to a is a collection of triangles that, when we forget the maps to aa, form a covering family of u0u_0 in CC. This is at the very least a coverage, and so we have a site.